Preamble

Improving MCUCN Code to Simulate Ultracold Neutron Storage and Transportation in Superfluid Helium

Xue-Fen Han,¹,²,³ Fei Shen,¹,² Bin Zhou,¹,² Xiao-Xiao Cai,¹,² Tian-Cheng Yi,¹,² Zhi-Liang Hu,¹,² Song-Lin Wang,¹,² Tian-Jiao Liang,¹,²,³,⁴,† and Robert Golub⁵

¹Institute of High Energy Physics, Chinese Academy of Sciences (CAS), Beijing 100049, China
²Spallation Neutron Source Science Center (SNSSC), Dongguan 523803, China
³University of Chinese Academy of Sciences, Beijing 100049, China
⁴Guangdong-Hong Kong-Macao Joint Laboratory for Neutron Scattering Science and Technology, Dongguan 523803, Guangdong, China
⁵Department of Physics, North Carolina State University, Raleigh, NC 27695, USA

The ultracold neutron (UCN) transport code MCUCN, designed initially for simulating UCN transportation from a solid deuterium (SD₂) source and neutron electric dipole moment experiments, could not accurately simulate UCN storage and transportation in a superfluid ⁴He (SFHe, He II) source. This limitation arose from the absence of a ⁴He upscattering mechanism and the absorption of ³He. Additionally, the source energy distribution provided in MCUCN differs from that in an SFHe source. This study introduced enhancements to MCUCN to address these constraints, explicitly incorporating the ⁴He upscattering effect, the absorption of ³He, losses caused by impurities on the converter wall, the UCN source energy distribution in SFHe, and transmission through materials with negative optical potential. Additionally, a Python-based visualization code for intermediate states and results was developed. To validate these enhancements, we systematically compared simulation results of the Lujan Center Mark3 UCN system from MCUCN and the improved MCUCN code (iMCUCN) with UCNtransport simulations. We also compared the results of the SUN1 system simulated by MCUCN and iMCUCN with measurement results. The study demonstrates that iMCUCN effectively simulates the storage and transportation of ultracold neutrons in He II.

Keywords: ultracold neutron, storage, transportation, improved MCUCN code, upscattering effect, absorption by ³He

Introduction

Ultracold neutrons (UCNs) possess kinetic energy less than 300 neV, equivalent to temperatures below 3 mK or wavelengths longer than 500 Å \cite{1}. Their low energy allows complete reflection and confinement in materials with high Fermi potential \cite{2,3}, facilitating easy storage and counting. Since their discovery in 1969 \cite{4,5}, UCNs have been pivotal in studying fundamental physics, encompassing the measurement of neutron electric dipole moment \cite{6}, neutron lifetime \cite{7}, beta decay \cite{8,9}, gravitational resonance spectroscopy \cite{10,11}, neutrino asymmetry \cite{12}, neutron-antineutron oscillations \cite{13}, and surface characterization of materials \cite{14}. These experiments hold promise for investigating CP-violating mechanisms beyond the Standard Model, the origin of baryon asymmetry in the universe \cite{15}, the neutron lifetime puzzle (a key parameter in Big Bang theory), the real neutron loss mechanism, the fifth force, dark matter, and so on.

Superfluid helium (SFHe) and solid deuterium (SD₂) are superthermal materials for producing UCNs through neutron inelastic scattering. Neutron absorption on deuteron limits the UCN lifetime in SD₂, necessitating a UCN buffer volume for storage. Conversely, UCNs can be stored in SFHe until they decay due to its zero absorption cross-section, making SFHe converters viable as experimental bottles \cite{16}. Additionally, the ideal UCN density in SFHe can be significantly higher than in SD₂ \cite{17}. While SD₂ UCN sources have seen substantial development over the last 40 years \cite{18-25} and are recognized as leading UCN sources globally \cite{26,27}, SFHe UCN sources are gaining prominence, with the CSNS team presently designing a He II-based UCN source \cite{28-32}.

To characterize and optimize UCN storage and transportation processes, Monte Carlo-based UCN codes such as PENTrack \cite{33}, GEANT4UCN \cite{34}, UCNtransport \cite{35}, and MCUCN \cite{36} are available. Despite sharing core physical models, each software has distinct merits: PENTrack offers a flexible geometry configuration interface, while GEANT4UCN includes simulations of neutron spins, protons, and electrons in electromagnetic fields. UCNtransport, while valuable, is not open-source. Conversely, MCUCN offers significant advantages by being open-source, highly extensible, and executable in parallel via shell scripts. However, it suffers from certain limitations when used to simulate SFHe UCN sources, including the absence of critical models that describe ⁴He upscattering, ³He absorption, and impurities on the converter wall. Furthermore, it lacks the capability to represent the source energy distribution for SFHe and simulate UCN transmission through materials with negative optical potential.

To enhance the MCUCN code, we performed several modifications, including: (1) incorporation of a ⁴He upscattering physical model, (2) inclusion of a ³He absorption model, (3) introduction of an impurities model, (4) improvements to the source energy distribution in SFHe and geometric description, (5) addition of transmission capabilities for materials with negative optical potential, and (6) development of visualization tools for simulating intermediate states and results. Section II outlines the physical models and flow of the iMCUCN code. Section III presents results simulated by MCUCN, iMCUCN, and UCNtransport. Section IV compares MCUCN and iMCUCN results with experimental data. Consequently, iMCUCN is deemed valid and promising for UCN storage and transportation simulations in He II.

II. Physical Model Enhancement and Flow of iMCUCN

The basic theory and key attributes of MCUCN are outlined in detail in reference \cite{36}. This code is a tool for Monte Carlo-based ultracold neutron transport simulations, focusing on the intricate interactions between UCNs and materials. However, certain limitations in its current form necessitate enhancements for a more comprehensive representation of UCN behavior in SFHe.

Currently, MCUCN does not account for the upscattering effect of ⁴He, the absorption of ³He, or losses due to impurities. Moreover, the code does not consider the UCN source energy distribution in SFHe. Furthermore, the omission of UCN transmission through materials with negative optical potential represents a notable gap that plays a pivotal role in the simulations detailed in Section III. Additionally, MCUCN lacks built-in visualization tools, leaving the task of developing suitable visualization mechanisms to the user. To address these limitations and enable more accurate simulation of UCN storage and transportation in SFHe, enhancements to MCUCN are imperative.

A. The Loss Mechanisms in SFHe

For He II, it is possible to formulate the expression for the ultracold neutron survival time in terms of various loss mechanisms \cite{37-39}:

$$\tau^{-1} = \tau^{-1}{\text{up}} + \tau^{-1}}} + \tau^{-1{\text{wall}} + \tau^{-1}} + \tau^{-1{\text{leakage}} + \tau^{-1}$$}} \tag{1

Here, the reciprocal of $\tau$ represents the UCN total loss rate. $\tau_{\text{up}}$ is influenced by the UCN upscattering effect in He II. $\tau_{\text{abs}}$ arises from the substantial neutron absorption cross-section of ³He. When a UCN encounters a wall coated with a material of high Fermi potential, losses may occur due to wall absorption and upscattering ($\tau_{\text{wall}}$). The parameter $\tau_{\beta} = 880$ s represents the neutron decay time. $\tau_{\text{leakage}}$ accounts for UCN leakage from gaps in the UCN valve and storage volume wall. $\tau_{\text{impurities}}$ represents losses caused by absorption and upscattering of impurities that deposit on the walls of the production volume \cite{39}. While MCUCN includes wall collisions, leakage, and decay models, it lacks these crucial additional loss mechanisms.

The upscattering rate of superfluid ⁴He, $\tau_{\text{up}}$, is temperature-dependent and can be divided into three parts \cite{40}: one-phonon absorption, two-phonon scattering, and roton-phonon scattering. H. Yoshiki summarized the formula for $\tau_{\text{up}}$ \cite{41}:

$$\tau_{\text{up}} = A T + BT^7 + CT^2 e^{-8.6} \tag{2}$$

where the first term originates from one-phonon absorption, the second from two-phonon scattering, and the third from roton-phonon scattering. Thus, the upscattering time constant is related to the He II temperature. According to measurements in Ref. \cite{42}, $A = 61.1$ s⁻¹, $B = 7.6 \times 10^{-3}$ s⁻¹K⁻⁷, and $C = 5.22$ s⁻¹K⁻⁷ for 1.6 K He II, resulting in $\tau_{\text{up}} = 3.4$ s. For $T = 0.7$ K, $A = 130$ s⁻¹ \cite{40,43}, $B = 9.0 \times 10^{-3}$ s⁻¹K⁻⁷, $C = 18$ s⁻¹K⁻⁷, leading to $\tau_{\text{up}} = 1247$ s. The UCN upscattering loss rate in He II has been experimentally confirmed in Refs. \cite{40,44} and plays a significant role at higher temperatures but a lesser role at lower temperatures.

According to Ref. \cite{38}, the wall loss time constant $\tau_{\text{wall}}$ is defined as:

$$\tau_{\text{wall}} \bar{\mu}(E_{\text{UCN}}) v S \tag{3}$$

where $\bar{\mu}(E_{\text{UCN}})$ is the average loss probability per bounce, dependent on UCN energy. It is noteworthy that higher-energy UCNs are more likely to be lost compared to lower-energy ones. Here, $V$ represents the converter volume, $S$ is its surface area, and $v$ denotes the neutron velocity in vacuum.

The absorption time constant $\tau_{\text{abs}}$ is given by:

$$= n_{\text{3He}}\sigma_a v = n_{\text{3He}}\sigma_a(\text{th})v(\text{th}) \tag{4}$$

This term remains independent of UCN velocity due to the $1/v$ principle of the ³He thermal neutron absorption cross-section $\sigma_a$. $\sigma_a$ is 5333 b at a velocity of 2200 m/s. The notation $n_{\text{3He}}$ denotes the number density of ³He atoms. The natural abundance ratio of $n_{\text{3He}}$ to $n_{\text{4He}}$, at $1.37 \times 10^{-6}$, corresponds to $3 \times 10^{16}$ cm⁻³ at 1 K and an absorption time constant of 0.028 s. This parameter remains crucial in the total loss rate calculation until the abundance of ³He is reduced to below $10^{-12}$.

The leakage time constant $\tau_{\text{leakage}}$ is defined as:

$$\tau_{\text{leakage}} S_{\text{leakage}} v \tag{5}$$

In comparison with $\tau_{\text{wall}}$, this term corresponds to wall loss with $\mu = 1$, where $S_{\text{leakage}}$ represents the total area of gaps that can leak UCNs.

B. Initial Energy Distribution in SFHe

The MCUCN code incorporates uniform and Gaussian energy distributions but does not include other forms of energy spectrum distribution. To simulate the SFHe UCN source, we integrated a Monte Carlo sampling function to generate the energy distribution that accurately represents the superfluid helium UCN source.

The relationship between the real space UCN density $n$ with energy $E$ in the range $(E, E + dE)$ and the phase space density $\rho$ is \cite{38}:

$$n(E)dE = = \rho d^3p \tag{6}$$

where $d^3r d^3p$ is the phase space volume and $d^3r$ is the real space volume. $N$ is the UCN number. When produced in SFHe, the UCNs fill the phase space with a constant density of $\rho$. Then:

$$n(E)dE \propto p^2dp \propto \tag{7}$$

Thus the initial UCN spectrum in the SFHe converter is:

$$dP_{\text{source}} \propto C. \tag{8}$$

C. The iMCUCN Code

As depicted in Fig. 1 [FIGURE:1], the iMCUCN code models UCN losses during transportation. The red sections delineate mechanisms for (a) generating the energy distribution of the UCN source in SFHe, (b) simulating the upscattering effect of ⁴He, (c) simulating absorption by residual ³He contamination, (d) simulating absorption and upscattering by impurities, and (e) simulating UCN transmission through materials with negative optical potential. Polypropylene (PP) possesses a negative optical potential and serves as the foil confining SFHe \cite{45}. The code MCUCN incorporates transmission physics through materials with positive optical potential but overlooks those with negative potential. Thus it is necessary to integrate the negative potential for PP foil, as described in the next section. The iMCUCN code also integrates geometry descriptions to simulate various volumes and includes a visualization component. UCNs are considered lost upon encountering the detector, the heat exchanger, or other components.

Fig. 1 showcases the iMCUCN's physical model, symbolized by diamonds \cite{36}. Absorption may result in the loss of UCNs produced in the deuterium converter, yet the absorption rate within the superfluid ⁴He UCN source is zero. Neutrons undergo continuous decay during simulations, with ⁴He upscattering effects and ³He absorption occurring solely in He II. Impurities are identified on the wall of the converter. UCNs encountering surfaces of the detector, heat exchanger, and other components are either lost or detected. Upon collision with a wall surface, UCNs will pass through if the wall is virtual, or they may be reflected, absorbed, or transmitted through the surface. Reflections manifest as either diffuse or specular, allowing reflected UCNs to persist and embark on subsequent collisions. Surface absorption arises from both absorption and inelastic scattering by the wall. The simulation concludes for a UCN upon reaching the exit window (i.e., the detector), marking the end of its journey and prompting the initiation of a new cycle with the generation of another UCN in the converter.

Fig. 1. (Color online) Illustration of the flow of physical models in the iMCUCN code \cite{36}. Diamonds indicate the physical models utilized for selecting a loss mechanism or proceeding to the exit window during transportation. Black characters represent original code sections, whereas our additions are highlighted in red.

III. Comparison of Simulations by MCUCN, iMCUCN, and UCNtransport Code for Lujan Center Mark3 UCN Transportation System

This section elucidates the extraction efficiency comparison of the proposed Lujan Center Mark3 UCN transportation system as simulated by MCUCN, iMCUCN, and UCNtransport. A horizontal near-foil geometry is constructed and its source and geometry are verified. The findings indicate that the variance in transmission rates simulated by UCNtransport \cite{46} and iMCUCN is less than 10% for the identical extraction system utilizing a He II source at 1.6 K. The discrepancies in transmission rates simulated by MCUCN are approximately twice those obtained from UCNtransport, mainly because MCUCN lacks an upscattering physical model in SFHe.

A. Basic Conditions

According to Ref. \cite{45}, an overview of the horizontal extraction geometry is shown in Fig. 2 [FIGURE:2]. 10,000 UCNs with energy below $U_{\text{58Ni}} - U_{\text{HeII}} = 316.5$ neV are produced isotropically and uniformly in a 40 L He cylinder converter. As mentioned above, $\tau_{\text{up}} = 3.4$ s for 1.6 K He II. When propagating through PP foil, the UCN transmission loss attributable to neutron elastic scattering off density inhomogeneities within the bulk is contingent upon both the macroscopic cross-section and the PP foil thickness \cite{46,47}. With an elastic scattering mean-free-path, $\lambda_{\text{scat}} = 20$ µm, for inhomogeneities, and the employed foil's thickness, $d = 30$ µm, the value of $\tilde{d} = 1.5$ was utilized as an input for the iMCUCN code.

The detector counts arriving UCNs with energies lower than $U_{\text{58Ni}}$ \cite{45}. The input parameters in the simulation are listed in Table 1. The angular distribution for diffuse reflection follows the Lambertian model \cite{38}. Gravity, loss due to wall absorption, upscattering effect in superfluid helium, and decay during UCN storage and transportation are included.

Fig. 2. (Color online) The horizontal near-foil geometry of Lujan center Mark3 UCN transportation system \cite{45}.

Table 1 [TABLE:1], sourced from Ref. \cite{45}. Input parameters for the MCUCN and iMCUCN code.

Configuration Parameters Generated UCN number 10,000 Detected UCN energy < 335 neV ($V_{\text{58Ni}}$) Cylinder $D = 0.36$ m, $L = 0.4$ m Radius of guide 0.09 m UCN Upscattering time 3.4 s (1.6 K) UCN decay time 880 s $V_{\text{58Ni}}$ 18.5 neV $\eta_{\text{58Ni}}$ 335 neV Diffuse rate of Al $5 \times 10^{-4}$ $\lambda_{\text{scat}}$ of idealized Al and PP 54 neV $\lambda_{\text{scat}}$ of PP $5.19 \times 10^{-5}$ -8 neV Thickness of Al and PP 20 µm Thickness of PP 30 µm Length of heat exchanger 30 µm Loss of foil support grid an 4 m guide loss to external volume 0.09 m

B. The Energy Distribution in SFHe

When UCNs are produced by inelastic scattering of 8.9 Å neutrons in superfluid helium, their angular distribution has a 6% asymmetry \cite{48}. This real but short-lived physical process can be disregarded, so the Monte Carlo method uniformly and isotropically generates the UCNs in He-II. The UCN energy distribution in He II in iMCUCN correctly follows Eq. 8, as shown in Fig. 3 [FIGURE:3]. It also presents the uniform energy distribution in MCUCN.

Fig. 3. (Color online) The normalized source UCN spectrum at different positions in the transport system. The red line stands for the formal expression $dP_{\text{source}}/dE \propto E$. The green line is the UCN spectrum distribution in the UCN converter. The blue line is the uniform UCN source distribution in MCUCN.

C. The Geometry of UCN Transport System

Extending the geometry code in MCUCN to accommodate various experiments or designs is crucial. Utilizing the developed Python code, we can assess the accuracy of the constructed geometry and identify any potential gaps. Fig. 4 [FIGURE:4] (a), (b), and (c) depict the simulation results, indicating the points where UCNs collide with the wall of the transport system. We projected the results into XOY, XOZ, and YOZ planes to verify whether the geometry is the one we aim to construct and to identify if there are any gaps. Fig. 4(d) to (f) provide the trajectories of UCNs on XOY, XOZ, and YOZ planes in the extraction process. The horizontal near-foil extraction geometry reveals no gaps, confirming the accuracy of the geometry model.

Fig. 4. (Color online) Figures (a) to (c) show the collision points when UCNs collide with the geometry during the transport process. Figures (d) to (f) show the trajectories of the first 100 UCNs.

D. Comparison of the Transmission Rates Simulated by MCUCN, iMCUCN, and UCNtransport

Tables 2 and 3 delineate the simulated outcomes by UCNtransport, MCUCN, and iMCUCN. Table 2 [TABLE:2] details the optimization process for the horizontal near-foil extraction system, changing simulation parameters based on the preceding step. A marked discrepancy in transmission rates is observed between MCUCN and UCNtransport simulations, whereas the variance between iMCUCN and UCNtransport remains under 10%.

Table 2. Comparison of transmission rates simulated by MCUCN, iMCUCN, and UCNtransport for steps to reach $\varepsilon_{\text{sim}} = 36\%$ based on the Lujan Center Mark3 horizontal near-foil extraction system. The differences ($D$) between the simulation results of iMCUCN and UCNtransport are listed.

Configuration $T_{\text{MCUCN}}$ (%) $T_{\text{iMCUCN}}$ (%) $T_{\text{UCNtransport}}$ (%) $D$ (%) Baseline: Al foil, $P_{\text{diffuse}} = 3\%$ 66.6 21.2 20.0 6.0 $P_{\text{diffuse}} = 50\%$ in converter 71.0 26.0 24.0 8.3 $P_{\text{diffuse}} = 50\%$ in vertical volume 71.0 27.0 25.0 8.0 Switch foil from Al to ideal PP 71.0 35.0 33.0 6.1 Add PP elastic scattering 71.0 33.0 32.0 3.1 Add PP foil support grid loss 64.0 30.0 29.0 3.4 Add 4 m guide to external volume 51.0 24.0 23.0 4.3

Table 3 [TABLE:3]. Comparison of the discrepancies between MCUCN, iMCUCN, and UCNtransport when changing the vertical column's height $h_{\text{column}}$ with $P_{\text{diffuse}} = 50\%$ on its side walls.

$h_{\text{column}}$ $T_{\text{MCUCN}}$ $T_{\text{iMCUCN}}$ $T_{\text{UCNtransport}}$ 0.09 m 71.0% 27.0% 25.0% 0.18 m 76.0% 31.0% 29.0% 0.36 m 81.0% 35.0% 33.0%

IV. Comparison of MCUCN, iMCUCN Simulated and SUN1 Experimental Results

To confirm the validity of the iMCUCN code, we performed simulations using MCUCN and iMCUCN and compared the results with the SUN1 measured results reported by Oliver Zimmer et al. \cite{39,49}. The analysis revealed that the disparities in count rates between the experimental measurements and iMCUCN simulations are small. In contrast, the predicted transmission rates by MCUCN exhibit deviations from the experimental results. This discrepancy predominantly arises from the lack of essential physical processes in MCUCN.

A. The Experimental Configuration and Parameters

Fig. 5 [FIGURE:5] illustrates the experimental configuration. The ⁴He converter exhibits a rectangular geometry measuring 100 cm × 7 cm × 7 cm, with Be coated on the front and rear faces and BeO used for the sides. The vertically oriented extraction guide is positioned at a height of 10 cm, with the shutter situated midway along its length. Following this, there is a 7 cm horizontal bend leading to a second vertical guide extending for 15 cm, ultimately leading to the detector at its end. Constructed from stainless steel, the guides have a diameter of 23 mm. The gap area at the shutter is 1.73 mm². Refer to Table 4 [TABLE:4] for detailed information on various input parameters.

The initial UCN spectrum follows Eq. 8, and the maximum energy is 261 neV as determined by the optical potential of BeO. The measured UCN count rate at the detector of the SUN1 system is depicted in Fig. 6 [FIGURE:6]. During a scenario where cold neutrons are incident while the shutter is closed, the detector detects UCNs attributed to the small gap between shutter and guide. The UCN count rates at the detector reached a saturation point after approximately 200 s. At around 800 s, the cold neutron beam was deactivated, coinciding with the shutter's opening to the UCN guide. A peak in the UCN count rate at the detector was observed after about 3 s. Subsequently, the shutter was closed after 135 s.

Fig. 5. The geometry of SUN1 \cite{39,49}.

Fig. 6 [FIGURE:6]. (Color online) SUN1 experimental results at detector \cite{39}. The red lines are fitted.

Table 4. Input parameters for the SUN1 experiment utilized in the simulation \cite{39,49}.

Configuration Parameters Converter dimensions 100 cm × 7 cm × 7 cm Radius of guide 0.0115 m $V_{\text{stainless}}$ 18.5 neV $\eta_{\text{stainless}}$ 261.0 neV Open shutter at $1.35 \times 10^{-5}$ Close shutter at 252.0 neV $5 \times 10^{-6}$ 183.0 neV $9.3 \times 10^{-5}$ 804 s 937 s

B. Scenario Before Opening Shutter

Before opening the shutter, the UCN number in the converter as a function of time is \cite{38}:

$$N(t) = \rho(t)V = VP\tau(\text{cid:16})1 - e^{-t}(\text{cid:17}) \tag{9}$$

where $\rho(t)$ is the UCN density in the converter, $V$ is the volume of the converter, $P$ is the UCN production rate per unit volume, and $\tau$ is the buildup time as defined in Eq. 1, representing the UCN survival time in the SFHe converter. The UCN number can reach a saturation point when the UCN generation time is sufficiently long, and the saturation UCN number in the converter will be $N_{\text{sat}} = VP\tau$ where $N_{\text{sat}}$ represents the saturation UCN number in the converter.

The UCN count rate at the detector in Figure 6 can be expressed as $C(t) \propto N(t)\tau_{\text{gap}}$ where $\tau_{\text{gap}}$ is determined by the gap area between the shutter and the guide before the shutter is opened and also follows Eq. 5.

In the converter, the $\beta$ decay term remains constant, and the absorption of ³He cannot be disregarded under this condition. Both the He-II upscattering and the abundance of ³He are independent of UCN velocity. According to Ref. \cite{39}, it was reported that $\tau(\text{experiment})^{-1} = 1/67$ s⁻¹, $\tau^{-1}{\text{abs}} = 1/315$ s⁻¹, $\tau^{-1}}} = 1/1247$ s⁻¹, $\tau^{-1{\text{wall}} = 1/278$ s⁻¹, $\tau^{-1}} = 1/880$ s⁻¹, and $\tau^{-1{\text{leakage}} + \tau^{-1} = 1/161$ s⁻¹. Both the impurities time constant and leakage time constant remain ambiguous. To simplify the simulation, we assume that the losses caused by impurities and leakage are energy-dependent, similar to wall loss.}

The leakage count rate at the detector is proportional to the area of the gap between the shutter and guide. The gap of the shutter, unspecified in Ref. \cite{39,49}, can only be determined by comparing with simulated results through changing the size of the gap. As illustrated in Fig. 7 [FIGURE:7], we observed an increase in the UCN count rate before opening the shutter with increasing shutter gap size because the leakage rate is proportional to the gap area. The gap whose simulation results coincide with the experimental results before opening the shutter is determined to be 1.73 mm².

Table 5 [TABLE:5]. The two decay time constants under different conditions.

Condition $d_g$(%) $d_c$(%) $\tau$(s) $\tau_1$(s) $\tau_2$(s) SUN1 in \cite{39} - - 67 13 34 iMCUCN 1.5 3.0 71.0 13.7 31.7 iMCUCN 1.0 3.0 71.0 15.0 33.0

Fig. 7 [FIGURE:7]. (Color online) Fit the shutter gap.

C. Scenario After Opening of the Shutter

After opening the shutter, the accumulated UCNs in the converter will be released. Neglecting gravity and the energy distribution of UCNs, and closing the shutter of cold neutron, the decay of UCNs in the converter after opening the shutter can be described by:

$$N(t) = N_{\text{sat}}e^{-\tau_{\text{guide}}} \tag{10}$$

where $\tau_{\text{guide}}$ is the guide leakage constant after opening the shutter. According to Eq. 11, the UCN count rate at the detector can be expressed as $C(t) \propto N(t)/\tau_{\text{guide}}$. In real scenarios, this process is dependent on the UCN energy in the converter. Thus the UCN number in the converter does not follow a strict single time constant decay. The UCN count rate at the detector before opening the shutter is nearly proportional to the total UCN number in the converter, and researchers \cite{39,49} utilize two time decay constants to fit the detected UCN count rate after opening the shutter. In the SUN1 experiment, the fitted two decay time constants are $(13 \pm 1)$ s and $(34 \pm 1)$ s.

The decay time after opening the shutter is primarily influenced by the area of the guide, which has been determined experimentally. It can also be affected by the unknown diffuse reflection probabilities of the guide ($d_g$) and converter ($d_c$). The diffuse reflection probability of polished stainless steel coated in the guide typically falls within the range of 0.5 m to 1 m. Therefore, we present the decay time constants for some results in Table 5.

The optimal condition corresponds to a converter diffuse reflection probability $d_c = 3.0\%$ and a guide diffuse reflection probability $d_g = 1.5\%$. The final buildup time is 71.0 s, and the two decay time constants are 13.7 s and 31.7 s. The comparison between the simulated results and the measured results under this condition is shown in Fig. 8 [FIGURE:8]. The simulation results from iMCUCN under this condition closely align with the experimental data. Specifically, UCNs with lower energy contribute to the second decay time constant $\tau_2$. The discrepancy ranging from 840 s to 937 s between SUN1 measurements and data simulated by iMCUCN may arise from several factors. Firstly, the UCN spectrum generated in simulation may differ from the actual conditions in the experiment. Secondly, errors may be present in the data reading process from \cite{39}. To clarify these discrepancies, further experiments need to be conducted.

Fig. 8 [FIGURE:8]. (Color online) The top figure compares the simulation results of iMCUCN with the measured data under the selected conditions. The middle figure is identical to the top figure but is presented in logarithmic coordinates. The figure below illustrates the ratio of the experimental data to the iMCUCN simulation data.

D. Discussion

To further enhance comprehension of the underlying physics discussed in this section, additional details will be presented below.

Fig. 9 [FIGURE:9]. (Color online) The figure compares the simulation results of iMCUCN and MCUCN with the measured data under the optimal condition. The data simulated by iMCUCN agrees with the measured data, while that of MCUCN shows discrepancies.

The results obtained with $d_c = 3.0\%$ and $d_g = 1.5\%$ as simulated by MCUCN exhibit discrepancies with the experimental data depicted in the upper panel of Fig. 9. In the MCUCN simulation, the buildup time is 153.0 s, and the decay time constants are 15.3 s and 38.3 s. In the absence of losses due to impurities, leakage, absorption of ³He, and upscattering in ⁴He, the time required to reach the saturation UCN count rate is longer, allowing more UCNs to survive and be extracted. According to Eq. 3, wall loss depends on UCN energy; i.e., the lower the UCN energy, the smaller the chance of being lost after collision with the wall. The velocity of low-energy UCN is slower and takes more time to reach the detector. Therefore, the slope of the UCN count rate at the detector after opening the shutter simulated by MCUCN is relatively smaller than that simulated by iMCUCN in the upper panel of Fig. 9.

Fig. 10 [FIGURE:10] depicts the UCN spectrum within the converter at various time points simulated by iMCUCN. The figure at the top shows the UCN spectrum in the converter simulated by iMCUCN before 800 s, depicting the change over time. Initially, at 20 s, the UCNs begin to accumulate within the converter. Approximately 300 s later, a saturation point is nearly reached, indicating that the rate of UCN accumulation has equalized with the rate of their loss. However, from 804 s onward, the UCN spectrum within the converter begins to decrease due to the opening of the shutter. It is noteworthy that the saturated UCN spectrum observed at 800 s does not strictly adhere to the expected distribution as described by Eq. 8. This deviation is attributed to the presence of energy-dependent loss mechanisms, including impurities, leakage, and interactions with the converter walls. The figure below displays the UCN spectrum in the converter simulated by iMCUCN after 800 s. At 810 s, a greater number of faster UCNs have been extracted compared to those with lower energy. By 900 s, the majority of the UCNs have been extracted, with only a small number of low-energy UCNs remaining in the converter. The optical potentials of BeO, Be, and stainless steel are 261 neV, 252 neV, and 183 neV, respectively. UCNs with energy lower than 183 neV can be more effectively constrained than those ranging from 183 neV to 252 neV, as some of them may escape from the stainless steel guide before the shutter. UCNs with energies between 252 neV and 261 neV will be less constrained due to the presence of Be surfaces.

The total UCN number within the converter can be illustrated through simulation, as visualized in Fig. 11 [FIGURE:11]. Notably, there exists a significant discrepancy between the MCUCN-simulated total UCN number and those obtained from the iMCUCN simulation within the converter due to the lack of loss models for impurities, leakage, absorption of ³He, and upscattering in ⁴He in MCUCN.

Fig. 10. (Color online) The top figure illustrates the UCN spectrum in the converter simulated by iMCUCN before 800 s, showing the change over time. The bottom figure represents the spectrum after 800 s.

Fig. 11 [FIGURE:11]. (Color online) The total UCN number in the converter as a function of time simulated by iMCUCN and MCUCN.

Fig. 12 [FIGURE:12] illustrates the UCN spectrum at the detector. Prior to opening the shutter, the spectrum at the detector remains low due to the small leakage rate from the gap and the low statistical UCN count rate. At about 807 s, the UCN spectrum is near the peak, indicating significant extraction of UCNs to the detector. Higher-energy UCNs dominate the spectrum during this period. Subsequently, after 33 s, the extracted UCN number notably decreases, with lower-energy UCNs becoming more prominent than higher-energy ones. At 870 s, only lower-energy UCNs remain. Finally, by 900 s, only a few lower-energy UCNs can be extracted and detected.

Fig. 12. (Color online) The detected UCN spectrum at detector as a function of time.

V. Conclusion

In this paper, we extended the capabilities of the MCUCN code by incorporating the upscattering effect of ⁴He, absorption of ³He in superfluid helium, upscattering and absorption of impurities, source energy distribution in SFHe, a transmission model for materials with negative optical potential, and improved geometry description. We thoroughly validated the source and geometry parameters. Subsequent analysis showed that the data simulated by iMCUCN were in close agreement with simulations conducted by UCNtransport for the Lujan Center Mark3 UCN transport system and the SUN1 experimental data. These results demonstrate that iMCUCN is effective for simulating UCN storage and transport in SFHe UCN sources.

VI. Acknowledgement

This work was supported by the National Key R&D Program of China (No. 2024YFE0110001), the National Natural Science Foundation of China (U1932219), and the Mobility Programme endorsed by the Joint Committee of the Sino-German Center (M-0728). The authors would like to thank Dr. Zsigmond Geza for help in understanding the details of the original MCUCN code, and Prof. Kent Leung for help in understanding the origin of the elastic scattering in the inhomogeneous foil.

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